264,346 research outputs found

    A Multi-Grid Iterative Method for Photoacoustic Tomography

    Get PDF
    Inspired by the recent advances on minimizing nonsmooth or bound-constrained convex functions on models using varying degrees of fidelity, we propose a line search multigrid (MG) method for full-wave iterative image reconstruction in photoacoustic tomography (PAT) in heterogeneous media. To compute the search direction at each iteration, we decide between the gradient at the target level, or alternatively an approximate error correction at a coarser level, relying on some predefined criteria. To incorporate absorption and dispersion, we derive the analytical adjoint directly from the first-order acoustic wave system. The effectiveness of the proposed method is tested on a total-variation penalized Iterative Shrinkage Thresholding algorithm (ISTA) and its accelerated variant (FISTA), which have been used in many studies of image reconstruction in PAT. The results show the great potential of the proposed method in improving speed of iterative image reconstruction

    A Second Order Godunov Method for Multidimensional Relativistic Magnetohydrodynamics

    Full text link
    We describe a new Godunov algorithm for relativistic magnetohydrodynamics (RMHD) that combines a simple, unsplit second order accurate integrator with the constrained transport (CT) method for enforcing the solenoidal constraint on the magnetic field. A variety of approximate Riemann solvers are implemented to compute the fluxes of the conserved variables. The methods are tested with a comprehensive suite of multidimensional problems. These tests have helped us develop a hierarchy of correction steps that are applied when the integration algorithm predicts unphysical states due to errors in the fluxes, or errors in the inversion between conserved and primitive variables. Although used exceedingly rarely, these corrections dramatically improve the stability of the algorithm. We present preliminary results from the application of these algorithms to two problems in RMHD: the propagation of supersonic magnetized jets, and the amplification of magnetic field by turbulence driven by the relativistic Kelvin-Helmholtz instability (KHI). Both of these applications reveal important differences between the results computed with Riemann solvers that adopt different approximations for the fluxes. For example, we show that use of Riemann solvers which include both contact and rotational discontinuities can increase the strength of the magnetic field within the cocoon by a factor of ten in simulations of RMHD jets, and can increase the spectral resolution of three-dimensional RMHD turbulence driven by the KHI by a factor of 2. This increase in accuracy far outweighs the associated increase in computational cost. Our RMHD scheme is publicly available as part of the Athena code.Comment: 75 pages, 28 figures, accepted for publication in ApJS. Version with high resolution figures available from http://jila.colorado.edu/~krb3u/Athena_SR/rmhd_method_paper.pd

    Status of the differential transformation method

    Full text link
    Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

    Assessment and calibration of the γ equation transition model for a wide range of Reynolds numbers at low Mach

    Get PDF
    The numerical simulation of flows over large-scale wind turbine blades without considering the transition from laminar to fully turbulent flow may result in incorrect estimates of the blade loads and performance. Thanks to its relative simplicity and promising results, the Local-Correlation based Transition Modelling concept represents a valid way to include transitional effects into practical CFD simulations. However, the model involves coefficients to be tuned to match the required application. In this paper, the γ-equation transition model is assessed and calibrated, for a wide range of Reynolds numbers at low Mach, as needed for wind turbine applications. Different airfoils are used to evaluate the original model and calibrate it, whereas a large-scale wind turbine blade is employed to show that the calibrated model can lead to reliable solution for complex three-dimensional flows. The calibrated model shows promising results for both two-dimensional and three-dimensional flows, even if cross-flow instabilities are neglected

    Numerical Homogenization of Heterogeneous Fractional Laplacians

    Get PDF
    In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional Laplacian is a nonlocal operator in its standard form, however the Caffarelli-Silvestre extension allows for a localization of the equations. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a sub-grid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Since the solution of the extended problem on the critical boundary is of main interest, we construct a projective quasi-interpolation that has both dd and d+1d+1 dimensional averages over subsets in the spirit of the Scott-Zhang operator. We show that this operator satisfies local stability and local approximation properties in weighted Sobolev spaces. We further show that we can obtain a greater rate of convergence for sufficient smooth forces, and utilizing a global L2L^2 projection on the critical boundary. We present some numerical examples, utilizing our projective quasi-interpolation in dimension 2+12+1 for analytic and heterogeneous cases to demonstrate the rates and effectiveness of the method

    Efficient cosmological parameter sampling using sparse grids

    Full text link
    We present a novel method to significantly speed up cosmological parameter sampling. The method relies on constructing an interpolation of the CMB-log-likelihood based on sparse grids, which is used as a shortcut for the likelihood-evaluation. We obtain excellent results over a large region in parameter space, comprising about 25 log-likelihoods around the peak, and we reproduce the one-dimensional projections of the likelihood almost perfectly. In speed and accuracy, our technique is competitive to existing approaches to accelerate parameter estimation based on polynomial interpolation or neural networks, while having some advantages over them. In our method, there is no danger of creating unphysical wiggles as it can be the case for polynomial fits of a high degree. Furthermore, we do not require a long training time as for neural networks, but the construction of the interpolation is determined by the time it takes to evaluate the likelihood at the sampling points, which can be parallelised to an arbitrary degree. Our approach is completely general, and it can adaptively exploit the properties of the underlying function. We can thus apply it to any problem where an accurate interpolation of a function is needed.Comment: Submitted to MNRAS, 13 pages, 13 figure
    corecore